That looks like the formula $\displaystyle \sqrt{n}(\hat{\theta}-\theta)\sim{N}(0,I_Y^{-1}(\theta))$ adapted to Bin(1,p) for n observations, or, $\displaystyle \sqrt{I_Y(\theta)}(\hat{\theta}-\theta)\sim{N}(0,1)$ for large n

So, does the former ($\displaystyle \sqrt{n}(\hat{\theta}-\theta)->N(0,I_Y^{-1}(\theta))$) comes from Slutsky theorem? I just never seen this name before, and I was also given the above assymptotic formula without explaining where it comes from.

Ah! then, that assymptotic formula with Fisher information in it looks just like a version of the Central Limit Theorem. (edited: but of course it would, because it originates from CLT, I just didn't see the proof so I didn't see that it does. Sorry for off-top).